On multi-level bases for elliptic boundary value problems (Q1919943)

A multi-level basis in different spline spaces that preconditions the linear system arising from a Galerkin discretization method of an elliptic boundary value problem of order \(2r\) is presented. In the well-organized paper they start to propose the problem and summarize the most general results of \textit{W. Dahmen} and \textit{A. Kunoth} [Numer. Math. 63, No. 3, 315-344 (1992; Zbl 0757.65031)] regarding the condition number. Section 2 gives their own result that the condition number is \(\kappa={\mathcal O}((n+1)^2)\) for the different spline spaces considered. In Section 3 the proofs are presented where the proof of theorem 5 gives a detailed description how to construct the locally supported basis. Only for one point, how to choose a minimal determining set of coefficients, they refer to earlier publications.